Gr. 4 Overview

CCGPS
Frameworks
Teacher Edition

Mathematics

Fourth Grade
Grade Level Overview

Georgia Department of Education
Common Core Georgia Performance Standards Framework
Fourth Grade Mathematics • Grade Level Overview

Grade Level Overview

TABLE OF CONTENTS

Curriculum Map and pacing Guide…………………………………………………………………

Unpacking the Standards

• Standards of Mathematical Practice………………………..……………...…………..……

• Content Standards………………………………………….………………………….……

Arc of Lesson/Math Instructional Framework………………………………………………..……

Unpacking a Task………………………………………………………………………………..…

Routines and Rituals………………………………………………………………………………..

General Questions for Teacher Use………………………………………………………………...

Questions for Teacher Reflection……………………………………………………….………….

Depth of Knowledge……………………………………………………………………….…….…

Depth and Rigor Statement…………………………………………………………………………

Additional Resources Available

• 3-5 Problem Solving Rubric……………………………………………………………….

• Literature Resources……………………………………………………………….……….

• Technology Links…………………………………………………………………………..

Resources Consulted

MATHEMATICS GRADE 4 Grade Level Overview
Dr. John D. Barge, State School Superintendent
April 2012 Page 2 of 62
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Common Core Georgia Performance Standards
Fourth Grade
Common Core Georgia Performance Standards: Curriculum Map

Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7 Unit 8

Whole Fraction Adding and Multiplying Fractions and Geometry Measurement Show What We
Numbers, Place Equivalents Subtracting and Dividing Decimals Know
Value and Fractions Fractions
Rounding in
Computation

MCC4.NBT.1 MCC4.NF.1 MCC4.NF.3 MCC4.NF.4 MCC4.NF.5 MCC4.G.1 MCC4.MD.1 ALL
MCC4.NBT.2 MCC4.NF.2 MCC4.NF.6 MCC4.G.2 MCC4.MD.2
MCC4.NBT.3 MCC4.OA.1 MCC4.NF.7 MCC4.G.3 MCC4.MD.3
MCC4.NBT.4 MCC4.OA.4 MCC4.MD.4
MCC4.NBT.5 MCC4.MD.5
MCC4.NBT.6 MCC4.MD.6
MCC4.OA.1 MCC4.MD.7
MCC4.OA.2
MCC4.OA.3
MCC4.OA.4
MCC4.OA.5

These units were written to build upon concepts from prior units, so later units contain tasks that depend upon the concepts addressed in earlier units.
All units will include the Mathematical Practices and indicate skills to maintain.
NOTE: Mathematical standards are interwoven and should be addressed throughout the year in as many different units and tasks as possible in order to stress the natural connections that exist among
mathematical topics.

Grades 3-5 Key: G= Geometry, MD=Measurement and Data, NBT= Number and Operations in Base Ten, NF = Number and Operations, Fractions, OA = Operations and Algebraic Thinking.

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STANDARDS FOR MATHEMATICAL PRACTICE

The Standards for Mathematical Practice describe varieties of expertise that mathematics
educators at all levels should seek to develop in their students.

These practices rest on important “processes and proficiencies” with longstanding importance
in mathematics education. The first of these are the NCTM process standards of problem solving,
reasoning and proof, communication, representation, and connections. The second are the
strands of mathematical proficiency specified in the National Research Council’s report Adding
It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of
mathematical concepts, operations and relations), procedural fluency (skill in carrying out
procedures flexibly, accurately, efficiently and appropriately), and productive disposition
(habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a
belief in diligence and one’s own efficacy).

Students are expected to:
1. Make sense of problems and persevere in solving them.
In fourth grade, students know that doing mathematics involves solving problems and discussing
how they solved them. Students explain to themselves the meaning of a problem and look for
ways to solve it. Fourth graders may use concrete objects or pictures to help them conceptualize
and solve problems. They may check their thinking by asking themselves, “Does this make
sense?” They listen to the strategies of others and will try different approaches. They often will
use another method to check their answers.

2. Reason abstractly and quantitatively.
Fourth graders should recognize that a number represents a specific quantity. They connect the
quantity to written symbols and create a logical representation of the problem at hand,
considering both the appropriate units involved and the meaning of quantities. They extend this
understanding from whole numbers to their work with fractions and decimals. Students write
simple expressions, record calculations with numbers, and represent or round numbers using
place value concepts.

3. Construct viable arguments and critique the reasoning of others.
In fourth grade, students may construct arguments using concrete referents, such as objects,
pictures, and drawings. They explain their thinking and make connections between models and
equations. They refine their mathematical communication skills as they participate in
mathematical discussions involving questions like “How did you get that?” and “Why is that
true?” They explain their thinking to others and respond to others’ thinking.

4. Model with mathematics.
Students experiment with representing problem situations in multiple ways including numbers,
words (mathematical language), drawing pictures, using objects, making a chart, list, or graph,
creating equations, etc. Students need opportunities to connect the different representations and
explain the connections. They should be able to use all of these representations as needed. Fourth
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graders should evaluate their results in the context of the situation and reflect on whether the
results make sense.

5. Use appropriate tools strategically.
Fourth graders consider the available tools (including estimation) when solving a mathematical
problem and decide when certain tools might be helpful. For instance, they may use graph paper
or a number line to represent and compare decimals and protractors to measure angles. They use
other measurement tools to understand the relative size of units within a system and express
measurements given in larger units in terms of smaller units.

6. Attend to precision.
As fourth graders develop their mathematical communication skills, they try to use clear and
precise language in their discussions with others and in their own reasoning. They are careful
about specifying units of measure and state the meaning of the symbols they choose. For
instance, they use appropriate labels when creating a line plot.

7. Look for and make use of structure.
In fourth grade, students look closely to discover a pattern or structure. For instance, students use
properties of operations to explain calculations (partial products model). They relate
representations of counting problems such as tree diagrams and arrays to the multiplication
principal of counting. They generate number or shape patterns that follow a given rule.

8. Look for and express regularity in repeated reasoning.
Students in fourth grade should notice repetitive actions in computation to make generalizations
Students use models to explain calculations and understand how algorithms work. They also use
models to examine patterns and generate their own algorithms. For example, students use visual
fraction models to write equivalent fractions.

***Mathematical Practices 1 and 6 should be evident in EVERY lesson***

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CONTENT STANDARDS

Operations and Algebraic Thinking
CLUSTER #1: USE THE FOUR OPERATIONS WITH WHOLE NUMBERS TO SOLVE
PROBLEMS.
Mathematically proficient students communicate precisely by engaging in discussion
about their reasoning using appropriate mathematical language. The terms students
should learn to use with increasing precision with this cluster are:
multiplication/multiply, division/divide, addition/add, subtraction/subtract, equations,
unknown, remainders, reasonableness, mental computation, estimation, rounding.

MCC.4.OA.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7
as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal
statements of multiplicative comparisons as multiplication equations.

A multiplicative comparison is a situation in which one quantity is multiplied by a
specified number to get another quantity (e.g., “a is n times as much as b”). Students
should be able to identify and verbalize which quantity is being multiplied and which
number tells how many times.
Students should be given opportunities to write and identify equations and statements for
multiplicative comparisons.

Examples:
5 x 8 = 40: Sally is five years old. Her mom is eight times older. How old is Sally’s
Mom?
5 x 5 = 25: Sally has five times as many pencils as Mary. If Sally has 5 pencils, how
many does Mary have?

MCC.4.OA.2 Multiply or divide to solve word problems involving multiplicative
comparison, e.g., by using drawings and equations with a symbol for the unknown number
to represent the problem, distinguishing multiplicative comparison from additive
comparison.

This standard calls for students to translate comparative situations into equations with an
unknown and solve. Students need many opportunities to solve contextual problems.
Refer Table 2, included at the end of this document, for more examples.
Examples:
• Unknown Product: A blue scarf costs $3. A red scarf costs 6 times as much. How
much does the red scarf cost? (3 × 6 = p)
• Group Size Unknown: A book costs $18. That is 3 times more than a DVD. How
much does a DVD cost? (18 ÷ p = 3 or 3 × p = 18)

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• Number of Groups Unknown: A red scarf costs $18. A blue scarf costs $6. How
many times as much does the red scarf cost compared to the blue scarf? (18 ÷ 6 = p or 6
× p = 18)
When distinguishing multiplicative comparison from additive comparison, students
should note the following.
• Additive comparisons focus on the difference between two quantities.
o For example, Deb has 3 apples and Karen has 5 apples. How many more apples
does Karen have?
o A simple way to remember this is, “How many more?”
• Multiplicative comparisons focus on comparing two quantities by showing that one
quantity is a specified number of times larger or smaller than the other.
o For example, Deb ran 3 miles. Karen ran 5 times as many miles as Deb. How
many miles did Karen run?
A simple way to remember this is “How many times as much?” or “How many times as
many?”

MCC.4.OA.3 Solve multistep word problems posed with whole numbers and having whole-
number answers using the four operations, including problems in which remainders must
be interpreted. Represent these problems using equations with a letter standing for the
unknown quantity. Assess the reasonableness of answers using mental computation and
estimation strategies including rounding.

The focus in this standard is to have students use and discuss various strategies. It refers to
estimation strategies, including using compatible numbers (numbers that sum to 10 or 100) or
rounding. Problems should be structured so that all acceptable estimation strategies will arrive at
a reasonable answer. Students need many opportunities solving multistep story problems using
all four operations.
Example 1:
On a vacation, your family travels 267 miles on the first day, 194 miles on the second day and 34
miles on the third day. How many miles did they travel total?
Some typical estimation strategies for this problem are shown below.
Student 1 Student 2 Student 3
I first thought about I first thought about 194. It I rounded 267 to
267 and 34. I noticed is really close to 200. I also 300. I rounded 194
that their sum is about have 2 hundreds in 267. to 200. I rounded
300. Then I knew that That gives me a total of 4 34 to 30. When I
194 is close to 200. hundreds. Then I have 67 in added 300, 200,
When I put 300 and 267 and the 34. When I put and 30, I know my
200 together, I get 67 and 34 together that is answer will be
500. really close to 100. When I about 530.
add that hundred to the 4
hundreds that I already had,
I end up with 500.

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The assessment of estimation strategies should only have one reasonable answer (500 or 530), or
a range (between 500 and 550). Problems will be structured so that all acceptable estimation
strategies will arrive at a reasonable answer.

Example 2:
Your class is collecting bottled water for a service project. The goal is to collect 300 bottles of
water. On the first day, Max brings in 3 packs with 6 bottles in each container. Sarah wheels in
6 packs with 6 bottles in each container. About how many bottles of water still need to be
collected?

Student 1 Student 2
First I multiplied 3 and 6 which First I multiplied 3 and 6 which
equals 18. Then I multiplied 6 and equals 18. Then I multiplied 6 and
6 which is 36. I know 18 plus 36 is 6 which is 36. I know 18 is about
about 50. I’m trying to get to 300. 20 and 36 is about 40. 40 + 20 =
50 plus another 50 is 100. Then I 60. 300 – 60 = 240, so we need
need 2 more hundreds. So we still about 240 more bottles.
need 250 bottles.

This standard references interpreting remainders. Remainders should be put into context for
interpretation. Ways to address remainders:
• Remain as a left over
• Partitioned into fractions or decimals
• Discarded leaving only the whole number answer
• Increase the whole number answer up one
• Round to the nearest whole number for an approximate result

Example:
Write different word problems involving 44 ÷ 6 = ? where the answers are best represented as:
• Problem A: 7
• Problem B: 7 r 2
• Problem C: 8
• Problem D: 7 or 8
• Problem E: 7 2/6

Possible solutions:
• Problem A: 7.
Mary had 44 pencils. Six pencils fit into each of her pencil pouches. How many pouches did she
fill? 44 ÷ 6 = p; p = 7 r 2. Mary can fill 7 pouches completely.

• Problem B: 7 r 2.

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Mary had 44 pencils. Six pencils fit into each of her pencil pouches. How many pouches could
she fill and how many pencils would she have left? 44 ÷ 6 = p; p = 7 r 2; Mary can fill 7
pouches and have 2 left over.

• Problem C: 8.
Mary had 44 pencils. Six pencils fit into each of her pencil pouches. What would the fewest
number of pouches she would need in order to hold all of her pencils? 44 ÷ 6 = p; p = 7 r 2;
Mary can needs 8 pouches to hold all of the pencils.

• Problem D: 7 or 8.
Mary had 44 pencils. She divided them equally among her friends before giving one of the
leftovers to each of her friends. How many pencils could her friends have received? 44 ÷ 6 = p;
p = 7 r 2; Some of her friends received 7 pencils. Two friends received 8 pencils.

• Problem E: 72/6.
Mary had 44 pencils and put six pencils in each pouch. What fraction represents the number of
pouches that Mary filled? 44 ÷ 6 = p; p = 72/6
Example:
There are 128 students going on a field trip. If each bus held 30 students, how many buses are
needed? (128 ÷ 30 = b; b = 4 R 8; They will need 5 buses because 4 busses would not hold all of
the students).
Students need to realize in problems, such as the example above, that an extra bus is needed for
the 8 students that are left over. Estimation skills include identifying when estimation is
appropriate, determining the level of accuracy needed, selecting the appropriate method of
estimation, and verifying solutions or determining the reasonableness of situations using various
estimation strategies. Estimation strategies include, but are not limited to the following.

• Front-end estimation with adjusting (Using the highest place value and estimating from the
front end, making adjustments to the estimate by taking into account the remaining amounts)

• Clustering around an average (When the values are close together an average value is selected
and multiplied by the number of values to determine an estimate.)

• Rounding and adjusting (Students round down or round up and then adjust their estimate
depending on how much the rounding affected the original values.)

• Using friendly or compatible numbers such as factors (Students seek to fit numbers together;
e.g., rounding to factors and grouping numbers together that have round sums like 100 or 1000.)

• Using benchmark numbers that are easy to compute (Students select close whole numbers for
fractions or decimals to determine an estimate.)

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CLUSTER #2: GAIN FAMILIARITY WITH FACTORS AND MULTIPLES.
should learn to use with increasing precision with this cluster are:
multiplication/multiply, division/divide, factor pairs, factor, multiple, prime, composite.

MCC.4.OA.4 Find all factor pairs for a whole number in the range 1–100. Recognize that a
whole number is a multiple of each of its factors. Determine whether a given whole number
in the range 1–100 is a multiple of a given one-digit number. Determine whether a given
whole number in the range 1–100 is prime or composite.

This standard requires students to demonstrate understanding of factors and multiples of
whole numbers. This standard also refers to prime and composite numbers. Prime
numbers have exactly two factors, the number one and their own number. For example,
the number 17 has the factors of 1 and 17. Composite numbers have more than two
factors. For example, 8 has the factors 1, 2, 4, and 8.

Common Misconceptions
A common misconception is that the number 1 is prime, when in fact; it is neither prime
nor composite. Another common misconception is that all prime numbers are odd
numbers. This is not true, since the number 2 has only 2 factors, 1 and 2, and is also an
even number.
When listing multiples of numbers, students may not list the number itself. Emphasize
that the smallest multiple is the number itself.
Some students may think that larger numbers have more factors. Having students share
all factor pairs and how they found them will clear up this misconception.

Prime vs. Composite:
• A prime number is a number greater than 1 that has only 2 factors, 1 and itself.
• Composite numbers have more than 2 factors.
Students investigate whether numbers are prime or composite by
• Building rectangles (arrays) with the given area and finding which numbers have more
than two rectangles (e.g., 7 can be made into only 2 rectangles, 1 × 7 and 7 × 1, therefore
it is a prime number).
• Finding factors of the number.
Students should understand the process of finding factor pairs so they can do this for any
number 1-100.
Example:
Factor pairs for 96: 1 and 96, 2 and 48, 3 and 32, 4 and 24, 6 and 16, 8 and 12.
Multiples can be thought of as the result of skip counting by each of the factors. When skip
counting, students should be able to identify the number of factors counted e.g., 5, 10, 15, 20
(there are 4 fives in 20).

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Example:
Factors of 24: 1, 2, 3, 4, 6,8, 12, 24
Multiples: 1, 2, 3, 4, 5, … , 24
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24
3, 6, 9, 12, 15, 15, 21, 24
4, 8, 12, 16, 20, 24
8, 16, 24
12, 24
24
To determine if a number between 1-100 is a multiple of a given one-digit number, some helpful
hints include the following:
• All even numbers are multiples of 2.
• All even numbers that can be halved twice (with a whole number result) are multiples of 4.
• All numbers ending in 0 or 5 are multiples of 5.

CLUSTER #3: GENERATE AND ANALYZE PATTERNS.
should learn to use with increasing precision with this cluster are: pattern (number or
shape), pattern rule.

MCC.4.OA.5 Generate a number or shape pattern that follows a given rule. Identify
apparent features of the pattern that were not explicit in the rule itself. For example, given
the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and
observe that the terms appear to alternate between odd and even numbers. Explain informally
why the numbers will continue to alternate in this way.
Patterns involving numbers or symbols either repeat or grow. Students need multiple
opportunities creating and extending number and shape patterns. Numerical patterns
allow students to reinforce facts and develop fluency with operations.
Patterns and rules are related. A pattern is a sequence that repeats the same process over
and over. A rule dictates what that process will look like. Students investigate different
patterns to find rules, identify features in the patterns, and justify the reason for those
features.
Example:
Pattern Rule Feature(s)
3, 8, 13, 18, 23,28, . Start with 3; add 5 The numbers alternately end with a 3 or an 8
5, 10, 15, 20, … Start with 5; add 5 The numbers are multiples of 5 and end with either 0
or 5. The numbers that 3nd with 5 are products of 5
and an odd number. The numbers that end in 0 are
products of 5 and an even number.

After students have identified rules and features from patterns, they need to generate a
numerical or shape pattern from a given rule.

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Example:
Rule: Starting at 1, create a pattern that starts at 1 and multiplies each number by 3. Stop
when you have 6 numbers.
Students write 1, 3, 9, 27, 8, 243. Students notice that all the numbers are odd and that
the sums of the digits of the 2 digit numbers are each 9. Some students might investigate
this beyond 6 numbers. Another feature to investigate is the patterns in the differences of
the numbers (3 – 1 = 2, 9 – 3 = 6, 27 – 9 = 18, etc.).
This standard calls for students to describe features of an arithmetic number pattern or
shape pattern by identifying the rule, and features that are not explicit in the rule. A t-
chart is a tool to help students see number patterns.

Example:
There are 4 beans in the jar. Each day 3 beans are added. How many beans are in the jar
for each of the first 5 days?
Day Operation Beans
0 3×0+4 4
1 3×1+4 7
2 3×2+4 10
3 3×3+4 13
4 3×4+4 16
5 3×5+4 19
Number and Operation in Base Ten
CLUSTER #1: GENERALIZE PLACE VALUE UNDERSTANDING FOR MULTI-
DIGIT WHOLE NUMBERS.
should learn to use with increasing precision with this cluster are: place value, greater
than, less than, equal to, ‹, ›, =, comparisons/compare, round.

MCC.4.NBT.1 Recognize that in a multi-digit whole number, a digit in one place represents
ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 =
10 by applying concepts of place value and division.

This standard calls for students to extend their understanding of place value related to
multiplying and dividing by multiples of 10. In this standard, students should reason
about the magnitude of digits in a number. Students should be given opportunities to
reason and analyze the relationships of numbers that they are working with.

Example:
How is the 2 in the number 582 similar to and different from the 2 in the number 528?
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MCC.4.NBT.2 Read and write multi-digit whole numbers using base-ten numerals,
number names, and expanded form. Compare two multi-digit numbers based on meanings
of the digits in each place, using >, =, and < symbols to record the results of comparisons.

This standard refers to various ways to write numbers. Students should have flexibility
with the different number forms. Traditional expanded form is 285 = 200 + 80 + 5.
Written form is two hundred eighty-five. However, students should have opportunities to
explore the idea that 285 could also be 28 tens plus 5 ones or 1 hundred, 18 tens, and 5
ones.

Students should also be able to compare two multi-digit whole numbers using appropriate
symbols.

MCC.4.NBT.3 Use place value understanding to round multi-digit whole numbers to any
place.

This standard refers to place value understanding, which extends beyond an algorithm or
procedure for rounding. The expectation is that students have a deep understanding of
place value and number sense and can explain and reason about the answers they get
when they round. Students should have numerous experiences using a number line and a
hundreds chart as tools to support their work with rounding.

Example:
Your class is collecting bottled water for a service project. The goal is to collect 300
bottles of water. On the first day, Max brings in 3 packs with 6 bottles in each container.
Sarah wheels in 6 packs with 6 bottles in each container. About how many bottles of
water still need to be collected?

Student 1 Student 2
First, I multiplied 3 and 6 First, I multiplied 3 and 6
which equals 18. Then I which equals 18. Then I
multiplied 6 and 6 which is multiplied 6 and 6 which is
36. I know 18 plus 36 is 36. I know 18 is about 20
about 50. I’m trying to get to and 36 is about 40. 40 + 20 =
300. 50 plus another 50 is 60. 300 – 60 = 2040, so we
100. Then I need 2 more need about 240 more bottles.
hundreds. So we still need
250 bottles.

Example:
On a vacation, your family travels 267 miles on the first day, 194 miles on the second day
and 34 miles on the third day. How many total miles did your family travel?
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Some typical estimation strategies for this problem:

I first thought about 276 I first thought about 194. I rounded 267 to 300. I
and 34. I noticed that their It is really close to 200. I rounded 194 to 200. I
sum is about 300. Then I also have 2 hundreds in rounded 34 to 30. When I
knew that 194 is close to 267. That gives me a total added 300, 200, and 30, I
200. When I put 300 and of 4 hundreds. Then I know my answer will be
200 together, I get 500. have 67 in 267 and the 34. about 530.
When I put 67 and 34
together that is really close
to 100. When I add that
hundred to the 4 hundreds
that I already had, I end up
with 500.

Example:
Round 368 to the nearest hundred.
This will either be 300 or 400, since those are the two hundreds before and after 368.
Draw a number line, subdivide it as much as necessary, and determine whether 368 is
closer to 300 or 400. Since 368 is closer to 400, this number should be rounded to 400.


There are several misconceptions students may have about writing numerals from verbal
descriptions. Numbers like one thousand do not cause a problem; however a number like one
thousand two causes problems for students. Many students will understand the 1000 and the 2
but then instead of placing the 2 in the ones place, students will write the numbers as they hear
them, 10002 (ten thousand two). There are multiple strategies that can be used to assist with this
concept, including place-value boxes and vertical-addition method.
Students often assume that the first digit of a multi-digit number indicates the "greatness" of a
number. The assumption is made that 954 is greater than 1002 because students are focusing on
the first digit instead of the number as a whole.
Students need to be aware of the greatest place value. In this example, there is one number with
the lead digit in the thousands and another number with its lead digit in the hundreds.

CLUSTER #2: USE PLACE VALUE UNDERSTANDING AND PROPERTIES OF
OPERATIONS TO PERFORM MULTI-DIGIT ARITHMETIC.
Students generalize their understanding of place value to 1,000,000, understanding the
relative sizes of numbers in each place. They apply their understanding of models for
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multiplication (equal-sized groups, arrays, area models), place value, and properties of
operations, in particular the distributive property, as they develop, discuss, and use
efficient, accurate, and generalizable methods to compute products of multi-digit whole
numbers. Depending on the numbers and the context, they select and accurately apply
appropriate methods to estimate or mentally calculate products. They develop fluency
with efficient procedures for multiplying whole numbers; understand and explain why the
procedures work based on place value and properties of operations; and use them to
solve problems. Students apply their understanding of models for division, place value,
properties of operations, and the relationship of division to multiplication as they
develop, discuss, and use efficient, accurate, and generalizable procedures to find
quotients involving multi-digit dividends. They select and accurately apply appropriate
methods to estimate and mentally calculate quotients, and interpret remainders based
upon the context. Mathematically proficient students communicate precisely by engaging
in discussion about their reasoning using appropriate mathematical language. The terms
students should learn to use with increasing precision with this cluster are: partition(ed),
fraction, unit fraction, equivalent, multiple, reason, denominator, numerator,
comparison/compare, ‹, ›, =, benchmark fraction.

MCC.4.NBT.4 Fluently add and subtract multi-digit whole numbers using the standard
algorithm.

Students build on their understanding of addition and subtraction, their use of place value
and their flexibility with multiple strategies to make sense of the standard algorithm.
They continue to use place value in describing and justifying the processes they use to
add and subtract.

This standard refers to fluency, which means accuracy, efficiency (using a reasonable
amount of steps and time), and flexibility (using a variety strategies such as the
distributive property). This is the first grade level in which students are expected to be
proficient at using the standard algorithm to add and subtract. However, other previously
learned strategies are still appropriate for students to use.

When students begin using the standard algorithm their explanation may be quite lengthy.
After much practice with using place value to justify their steps, they will develop
fluency with the algorithm. Students should be able to explain why the algorithm works.
Example: 3892
+ 1567
Student explanation for this problem:
1. Two ones plus seven ones is nine ones.
2. Nine tens plus six tens is 15 tens.
3. I am going to write down five tens and think of the10 tens as one more hundred.(Denotes
with a 1 above the hundreds column)
4. Eight hundreds plus five hundreds plus the extra hundred from adding the tens is 14
hundreds.
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5. I am going to write the four hundreds and think of the 10 hundreds as one more 1000.
(Denotes with a 1 above the thousands column)
6. Three thousands plus one thousand plus the extra thousand from the hundreds is five
thousand.

Example: 3546
− 928
Student explanations for this problem:
1. There are not enough ones to take 8 ones from 6 ones so I have to use one ten as 10 ones.
Now I have 3 tens and 16 ones. (Marks through the 4 and notates with a 3 above the 4
and writes a 1 above the ones column to be represented as 16 ones.)
2. Sixteen ones minus 8 ones is 8 ones. (Writes an 8 in the ones column of answer.)
3. Three tens minus 2 tens is one ten. (Writes a 1 in the tens column of answer.)
4. There are not enough hundreds to take 9 hundreds from 5 hundreds so I have to use one
thousand as 10 hundreds. (Marks through the 3 and notates with a 2 above it. Writes
down a 1 above the hundreds column.) Now I have 2 thousand and 15 hundreds.
5. Fifteen hundreds minus 9 hundreds is 6 hundreds. (Writes a 6 in the hundreds column of
the answer.)
6. I have 2 thousands left since I did not have to take away any thousands. (Writes 2 in the
thousands place of answer.)
Students should know that it is mathematically possible to subtract a larger number from
a smaller number but that their work with whole numbers does not allow this as the
difference would result in a negative number.

MCC.4.NBT.5 Multiply a whole number of up to four digits by a one-digit whole number,
and multiply two two-digit numbers, using strategies based on place value and the
properties of operations. Illustrate and explain the calculation by using equations,
rectangular arrays, and/or area models.

Students who develop flexibility in breaking numbers apart have a better understanding
of the importance of place value and the distributive property in multi-digit
multiplication. Students use base ten blocks, area models, partitioning, compensation
strategies, etc. when multiplying whole numbers and use words and diagrams to explain
their thinking. They use the terms factor and product when communicating their
reasoning. Multiple strategies enable students to develop fluency with multiplication and
transfer that understanding to division. Use of the standard algorithm for
multiplication is an expectation in the 5th grade.

This standard calls for students to multiply numbers using a variety of strategies.

Example:
There are 25 dozen cookies in the bakery. What is the total number of cookies at the
bakery?

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25 × 12 25 × 12 25 × 12
I broke 12 up into I broke 25 into 5 I doubled 25 and
10 and 2. groups of 5. cut 12 in half to get
25 × 10 = 250 5 × 12 = 60 50 × 6.
25 × 2 = 50 I have 5 groups of 50 × 6 = 300
250 + 50 = 300 5 in 25.
60 × 5 = 300

Example:
What would an array area model of 74 x 38 look like?

Examples:
To illustrate 154 × 6, students use base 10 blocks or use drawings to show 154 six times. Seeing
154 six times will lead them to understand the distributive property,
154 × 6 = (100 + 50 + 4) × 6
= (100 × 6) + (50 × 6) + (4 × 6)
=600 + 300 + 24 = 924.

The area model below shows the partial products for 14 x 16 = 224.
Using the area model, students first verbalize their understanding:
• 10 × 10 is 100
• 4 × 10 is 40
• 10 × 6 is 60, and
• 4 × 6 is 24.

Students use different strategies to record this type of thinking.

Students explain this strategy and the one below with base 10 blocks, drawings, or numbers.
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25
× 24
400 (20 × 20)
100 (20 × 5)
80 (4 × 20)
20 (4 × 5)
600

MCC.4.NBT.6 Find whole-number quotients and remainders with up to four-digit
dividends and one-digit divisors, using strategies based on place value, the properties of
operations, and/or the relationship between multiplication and division. Illustrate and
explain the calculation by using equations, rectangular arrays, and/or area models.

In fourth grade, students build on their third grade work with division within 100. Students need
opportunities to develop their understandings by using problems in and out of context.
Example:
A 4th grade teacher bought 4 new pencil boxes. She has 260 pencils. She wants to put the pencils
in the boxes so that each box has the same number of pencils. How many pencils will there be in
each box?
• Using Base 10 Blocks: Students build 260 with base 10 blocks and distribute them into 4
equal groups. Some students may need to trade the 2 hundreds for tens but others may
easily recognize that 200 divided by 4 is 50.
• Using Place Value: 260 ÷ 4 = (200 ÷ 4) + (60 ÷ 4)
• Using Multiplication: 4 × 50 = 200, 4 × 10 = 40, 4 × 5 = 20; 50 + 10 + 5 = 65; so 260 ÷ 4
= 65
This standard calls for students to explore division through various strategies.
Example:
There are 592 students participating in Field Day. They are put into teams of 8 for the
competition. How many teams get created?
592 divided by 8 592 divided by 8 I want to get to 592.
There are 70 eights in 560. I know that 10 eights is 80. 8 × 25 = 200
592 – 560 = 32 If I take out 50 eights that is 8 × 25 = 200
There are 4 eights in 32. 400. 8 × 25 = 200
70 + 4 = 74 592 − 400 = 192 200 + 200 + 200 = 600
I can take out 20 more eights 600 – 8 = 592
which is 160. I had 75 groups of 8 and took
192 − 160 = 32 one away, so there are 74
8 goes into 32 four times. teams.
I have none left. I took out
50, then 20 more, then 4
more. That’s 74.

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Example:
Using an Open Array or Area Model
After developing an understanding of using arrays to divide, students begin to use a more
abstract model for division. This model connects to a recording process that will be
formalized in the 5th grade.
1. 150 ÷ 6

Students make a rectangle and write 6 on one of its sides. They express their
understanding that they need to think of the rectangle as representing a total of 150.
1. Students think, “6 times what number is a number close to 150?” They recognize
that 6 × 10 is 60 so they record 10 as a factor and partition the rectangle into 2
rectangles and label the area aligned to the factor of 10 with 60. They express that
they have only used 60 of the 150 so they have 90 left.
2. Recognizing that there is another 60 in what is left, they repeat the process above.
They express that they have used 120 of the 150 so they have 30 left.
3. Knowing that 6 × 5 is 30, they write 30 in the bottom area of the rectangle and
record 5 as a factor.
4. Student express their calculations in various ways:
a. 150
−60 (6 × 10) 150 ÷ 6 = 10 + 10 + 5 = 25
90
−60 (6 × 10)
30
−30 (6 × 5)
0

b. 150 ÷ 6 = (60 ÷ 6) + (60 ÷ 6) + (30 ÷ 6) = 10 + 10 + 5 = 25

2. 1917 ÷ 9

A student’s description of his or her thinking may be:
I need to find out how many 9s are in 1917. I know that 200 x 9 is 1800.
So if I use 1800 of the 1917, I have 117 left. I know that 9 x 10 is 90. So if
I have 10 more 9s, I will have 27 left. I can make 3 more 9s. I have 200
nines, 10 nines and 3 nines. So I made 213 nines. 1917 ÷ 9 = 213.

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Often students mix up when to 'carry' and when to 'borrow'. Also students often do not notice the
need of borrowing and just take the smaller digit from the larger one. Emphasize place value and
the meaning of each of the digits.

Number and Operations- Fractions
CLUSTER #1: EXTEND UNDERSTANDING OF FRACTION EQUIVALENCE AND
ORDERING.
Students develop understanding of fraction equivalence and operations with fractions.
They recognize that two different fractions can be equal (e.g., 15/9 = 5/3), and they
develop methods for generating and recognizing equivalent fractions. Mathematically
proficient students communicate precisely by engaging in discussion about their
reasoning using appropriate mathematical language. The terms students should learn to
use with increasing precision with this cluster are: partition(ed), fraction, unit fraction,
equivalent, multiple, reason, denominator, numerator, comparison/compare, ‹, ›, =,
benchmark fraction.
MCC.4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using
visual fraction models, with attention to how the number and size of the parts differ even
though the two fractions themselves are the same size. Use this principle to recognize and
generate equivalent fractions.
This standard refers to visual fraction models. This includes area models, number lines or
it could be a collection/set model. This standard extends the work in third grade by using
additional denominators (5, 10, 12, and 100).
This standard addresses equivalent fractions by examining the idea that equivalent
fractions can be created by multiplying both the numerator and denominator by the same
number or by dividing a shaded region into various parts.
Example:

Technology Connection: http://illuminations.nctm.org/activitydetail.aspx?id=80

MCC.4.NF.2 Compare two fractions with different numerators and different
denominators, e.g., by creating common denominators or numerators, or by comparing to
a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two

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fractions refer to the same whole. Record the results of comparisons with symbols >, =, or
<, and justify the conclusions, e.g., by using a visual fraction model.
This standard calls students to compare fractions by creating visual fraction models or finding
common denominators or numerators. Students’ experiences should focus on visual fraction
models rather than algorithms. When tested, models may or may not be included. Students
should learn to draw fraction models to help them compare. Students must also recognize that
they must consider the size of the whole when comparing fractions (i.e., 1/2 and 1/8 of two
medium pizzas is very different from 1/2 of one medium and 1/8 of one large).

Example:
Use patterns blocks.
1. If a red trapezoid is one whole, which block shows 1/3?
2. If the blue rhombus is 1/3, which block shows one whole?
3. If the red trapezoid is one whole, which block shows 2/3?

Example:
Mary used a 12 × 12 grid to represent 1 and Janet used a 10 × 10 grid to represent 1. Each girl
shaded grid squares to show ¼. How many grid squares did Mary shade? How many grid
squares did Janet shade? Why did they need to shade different numbers of grid squares?
Possible solution: Mary shaded 36 grid squares; Janet shaded 25 grid squares. The total number
of little squares is different in the two grids, so ¼ of each total number is different.

Example:
There are two cakes on the counter that are the same size. The first cake has 1/2 of it left. The
second cake has 5/12 left. Which cake has more left?

Student 1: Area Model
The first cake has more left over. The second cake
has 5/12 left which is smaller than 1/2.

Student 2: Number Line Model
The first cake has more left over: 1/2 is bigger than

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5/12.

Student 3: Verbal Explanation
I know that 6/12 equals 1/2, and 5/12 is less than 1/2. Therefore, the second cake
has less left over than the first cake. The first cake has more left over.

Example:
ଵ ସ ହ
When using the benchmark of ଶ to compare to଺ and ଼, you could use diagrams such as
these:

ସ ଵ ଵ ହ ଵ ଵ ଵ ଵ ସ
଺
is ଺ larger than ଶ, while ଼ is ଼ larger than ଶ. Since ଺ is greater than ଼, ଺ is the greater
fraction.

Students think that when generating equivalent fractions they need to multiply or divide either
the numerator or denominator, such as, changing 12 to sixths. They would multiply the
denominator by 3 to get 16, instead of multiplying the numerator by 3 also. Their focus is only
on the multiple of the denominator, not the whole fraction.
Students need to use a fraction in the form of one such as 33 so that the numerator and
denominator do not contain the original numerator or denominator.
CLUSTER #2: BUILD FRACTIONS FROM UNIT FRACTIONS BY APPLYING AND
EXTENDING PREVIOUS UNDERSTANDINGS OF OPERATIONS ON WHOLE
NUMBERS.
Students extend previous understandings about how fractions are built from unit
fractions, composing fractions from unit fractions, decomposing fractions into unit
fractions, and using the meaning of fractions and the meaning of multiplication to
multiply a fraction by a whole number. Mathematically proficient students communicate
precisely by engaging in discussion about their reasoning using appropriate
mathematical language. The terms students should learn to use with increasing precision
with this cluster are: operations, addition/joining, subtraction/separating, fraction, unit
fraction, equivalent, multiple, reason, denominator, numerator, decomposing, mixed
number, rules about how numbers work (properties), multiply, multiple.

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MCC.4.NF.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
a. Understand addition and subtraction of fractions as joining and separating parts
referring to the same whole.
A fraction with a numerator of one is called a unit fraction. When students investigate fractions
other than unit fractions, such as 2/3, they should be able to join (compose) or separate
(decompose) the fractions of the same whole.
ଶ ଵ ଵ
Example: ଷ
ൌ ଷ
൅ ଷ

Being able to visualize this decomposition into unit fractions helps students when adding or
subtracting fractions. Students need multiple opportunities to work with mixed numbers and be
able to decompose them in more than one way. Students may use visual models to help develop
this understanding.
ଵ ଷ ସ ଵ ହ ହ ଷ ଶ ଵ
Example: 1 ସ – ସ
ൌ? → ସ
൅
ସ
ൌସ → ସ
െ
ସ
ൌ
ସ
‫ݎ݋‬
ଶ

Example of word problem:
ଷ ଶ
Mary and Lacey decide to share a pizza. Mary ate ଺ and Lacey ate ଺ of the pizza. How much of
the pizza did the girls eat together?
ଷ ଵ ଵ ଵ
Possible solution: The amount of pizza Mary ate can be thought of a ଺ or ଺ + ଺ + ଺. The amount
ଵ ଵ ଵ ଵ ଵ ଵ ଵ
of pizza Lacey ate can be thought of a ଺ + ଺. The total amount of pizza they ate is ଺+ ଺+ ଺+ ଺+ ଺or
ହ
଺
ofthe pizza. A fraction with a numerator of one is called a unit fraction. When students
investigate fractions other than unit fractions, such as 2/3, they should be able to join (compose)
or separate (decompose) the fractions of the same whole.
ଶ ଵ ଵ
Example: ଷ
ൌ
ଷ
൅
ଷ

Being able to visualize this decomposition into unit fractions helps students when adding or
subtracting fractions. Students need multiple opportunities to work with mixed numbers
and be able to decompose them in more than one way. Students may use visual models to
help develop this understanding.
ଵ ଷ ସ ଵ ହ ହ ଷ ଶ ଵ
Example: 1 ସ – ସ
ൌ? → ସ
൅ ସ
ൌସ → ସ
െ ସ
ൌ ସ
‫ݎ݋‬ ଶ

Example of word problem:
ଷ ଶ
Mary and Lacey decide to share a pizza. Mary ate ଺ and Lacey ate ଺ of the pizza. How
much of the pizza did the girls eat together?
ଷ ଵ ଵ ଵ
Possible solution: The amount of pizza Mary ate can be thought of a ଺ or ଺ + ଺ + ଺. The amount
ଵ ଵ ଵ ଵ ଵ ଵ ଵ
of pizza Lacey ate can be thought of a ଺ + ଺. The total amount of pizza they ate is ଺+ ଺+ ଺+ ଺+ ଺or
ହ
଺
of the pizza.
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b. Decompose a fraction into a sum of fractions with the same denominator in more than
one way, recording each decomposition by an equation. Justify decompositions, e.g., by
using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1
+ 1/8 = 8/8 + 8/8 + 1/8.
Students should justify their breaking apart (decomposing) of fractions using visual fraction
models. The concept of turning mixed numbers into improper fractions needs to be emphasized
using visual fraction models.
Example:

c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed
number with an equivalent fraction, and/or by using properties of operations and the
relationship between addition and subtraction.
A separate algorithm for mixed numbers in addition and subtraction is not necessary. Students
will tend to add or subtract the whole numbers first and then work with the fractions using the
same strategies they have applied to problems that contained only fractions.
Example:
ଷ ଵ
Susan and Maria need 8 ଼ feet of ribbon to package gift baskets. Susan has 3 ଼ feet of ribbon and
ଷ
Maria has 5 ଼ feet of ribbon. How much ribbon do they have altogether? Will it be enough to
complete the project? Explain why or why not.
The student thinks: I can add the ribbon Susan has to the ribbon Maria has to find out how much
ଵ ଷ
ribbon they have altogether. Susan has 3 ଼ feet of ribbon and Maria has 5 ଼ feet of ribbon. I can
ଵ ଷ ଵ
write this as 3 ଼ ൅ 5 ଼. I know they have 8 feet of ribbon by adding the 3 and 5. They also have ଼

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ଷ ସ ସ ସ
and ଼ which makes a total of ଼ more. Altogether they have 8 ଼ feet of ribbon. 8 ଼8 is larger than
ଷ
8 ଼ so they will have enough ribbon to complete the project. They will even have a little extra
ଵ
ribbon left: ଼
foot.

Example:
ଵ
Trevor has 4 ଼ pizzas left over from his soccer party. After giving some pizza to his friend, he has
ସ
2 ଼ of a pizza left. How much pizza did Trevor give to his friend?

ଵ ଷଷ
Possible solution: Trevor had 4 ଼ pizzas to start. This is ଼
of a pizza. The x’s show the pizza he
ସ ଶ଴
has left which is 2 ଼ pizzas or ଼
pizzas. The shaded rectangles without the x’s are the pizza he
ଵଷ ହ
gave to his friend which is ଼
or 1 ଼ pizzas.

Mixed numbers are introduced for the first time in 4th Grade. Students should have ample
experiences of adding and subtracting mixed numbers where they work with mixed numbers or
convert mixed numbers into improper fractions.
Example:
ଷ ଵ
While solving the problem, 3 ସ ൅ 2 ସ, students could do the following:

ଷ ଵ
Student 1: 3 + 2 = 5 and ସ ൅ ସ
ൌ 1, so 5 + 1 = 6.

ଷ ଷ ଷ ଵ
Student 2: 3 ସ ൅ 2 ൌ 5 ସ, so 5 ସ ൅ ൌ 6.
ସ

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ଷ ଵହ ଵ ଽ ଵହ ଽ ଶସ
Student 3: 3ସ ൌ ସ
and 2 ସ ൌ ସ, so ସ
൅ ସ
ൌ ସ
ൌ 6.

d. Solve word problems involving addition and subtraction of fractions referring to the
same whole and having like denominators, e.g., by using visual fraction models and
equations to represent the problem.
Example:
ଷ ଵ ଶ
A cake recipe calls for you to use ସ cup of milk, ସ cup of oil, and ସ cup of water. How much
liquid was needed to make the cake?

MCC.4.NF.4 Apply and extend previous understandings of multiplication to multiply a
fraction by a whole number.
a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to
represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).

This standard builds on students’ work of adding fractions and extending that work into multiplication.
ଷ ଵ ଵ ଵ ଵ
Example: ଺ ൌ ଺ ൅ ଺ ൅ ଺ ൌ 3 ൈ ଺
Number line:

Area model:

b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a
fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 ×
(1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)
ଶ ଶ
This standard extended the idea of multiplication as repeated addition. For example, 3 ൈ ହ
ൌହ൅
ଶ ଶ ଺ ଵ
൅ ൌ ൌ 6 ൈ ହ.
ହ ହ ହ

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Students are expected to use and create visual fraction models to multiply a whole number by a
fraction.

c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by
using visual fraction models and equations to represent the problem. For example, if each
person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party,
how many pounds of roast beef will be needed? Between what two whole numbers does your
answer lie?

This standard calls for students to use visual fraction models to solve word problems related to
multiplying a whole number by a fraction.
Example:
In a relay race, each runner runs ½ of a lap. If there are 4 team members how long is the race?
Student 1 – Draws a number line showing 4 jumps of ½:

Student 2 – Draws an area model showing 4 pieces of ½
joined together to equal 2:

Student 3 – Draws an area model representing 4 × ½ on a
grid, dividing one row into ½ to represent the multiplier:

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Example:
ଵ ଵ
Heather bought 12 plums and ate ଷ of them. Paul bought 12 plums and ate ସ of them. Which
statement is true? Draw a model to explain your reasoning.
a. Heather and Paul ate the same number of plums.
b. Heather ate 4 plums and Paul ate 3 plums.
c. Heather ate 3 plums and Paul ate 4 plums.
d. Heather had 9 plums remaining.

Examples:
Students need many opportunities to work with problems in context to understand the
connections between models and corresponding equations. Contexts involving a whole number
times a fraction lend themselves to modeling and examining patterns.
ଶ ଵ ଺
1. 3 ൈ ହ ൌ 6 ൈ ହ
ൌ ହ

ଷ
2. If each person at a party eats ଼ of a pound of roast beef, and there are 5 people at the party, how
many pounds of roast beef are needed? Between what two whole numbers does your answer lie?
A student may build a fraction model to represent this problem:


Students think that it does not matter which model to use when finding the sum or difference of
fractions. They may represent one fraction with a rectangle and the other fraction with a circle.
They need to know that the models need to represent the same whole.

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CLUSTER #4: UNDERSTAND DECIMAL NOTATION FOR FRACTIONS, AND
COMPARE DECIMAL FRACTIONS.
should learn to use with increasing precision with this cluster are: fraction, numerator,
denominator, equivalent, reasoning, decimals, tenths, hundreds, multiplication,
comparisons/compare, ‹, ›, =.
MCC.4.NF.5 Express a fraction with denominator 10 as an equivalent fraction with
denominator 100, and use this technique to add two fractions with respective denominators
10 and 100.1 For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.
This standard continues the work of equivalent fractions by having students change
fractions with a 10 in the denominator into equivalent fractions that have a 100 in the
denominator. In order to prepare for work with decimals (CCGPS.4.NF.6 and
CCGPS.4.NF.7), experiences that allow students to shade decimal grids (10×10 grids)
can support this work. Student experiences should focus on working with grids rather
than algorithms. Students can also use base ten blocks and other place value models to
explore the relationship between fractions with denominators of 10 and denominators of
100.
This work in 4th grade lays the foundation for performing operations with decimal
numbers in 5th grade.

Example:

Example:
Represent 3 tenths and 30 hundredths on the models below.
Tenths circle Hundredths circle

1
Students who can generate equivalent fractions can develop strategies for adding fractions with unlike
denominators in general. But addition and subtraction with unlike denominators in general is not a requirement for
this grade.
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MCC.4.NF.6 Use decimal notation for fractions with denominators 10 or 100. For example,
rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line
diagram.
Decimals are introduced for the first time. Students should have ample opportunities to
explore and reason about the idea that a number can be represented as both a fraction and
a decimal.
Students make connections between fractions with denominators of 10 and 100 and the
ଷଶ
place value chart. By reading fraction names, students say ଵ଴଴ as thirty-two hundredths
and rewrite this as 0.32 or represent it on a place value model as shown on the following
page.

Hundreds Tens Ones • Tenths Hundredths

• 3 2
ଷଶ
Students use the representations explored in MCC.4.NF.5 to understand ଵ଴଴ can be
ଷ ଶ ଷଶ
expanded to ଵ଴ and ଵ଴଴. Students represent values such as 0.32 or ଵ଴଴ on a number line.
ଷଶ ଷ଴ ଷ ସ଴ ସ ଷ଴
ଵ଴଴
is more than ଵ଴଴ (or ଵ଴) and less than ଵ଴଴ (or ଵ଴). It is closer to ଵ଴଴ so it would be placed
on the number line near that value.

MCC.4.NF.7 Compare two decimals to hundredths by reasoning about their size.
Recognize that comparisons are valid only when the two decimals refer to the same whole.
Record the results of comparisons with the symbols >, =, or <, and justify the conclusions,
e.g., by using a visual model.

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Students should reason that comparisons are only valid when they refer to the same
whole. Visual models include area models, decimal grids, decimal circles, number lines,
and meter sticks.
Students build area and other models to compare decimals. Through these experiences
and their work with fraction models, they build the understanding that comparisons
between decimals or fractions are only valid when the whole is the same for both cases.
Each of the models below shows 3/10 but the whole on the right is much bigger than the
whole on the left. They are both 3/10 but the model on the right is a much larger quantity
than the model on the left.
When the wholes are the same, the decimals or fractions can be compared.
Example:
Draw a model to show that 0.3 < 0.5. (Students would sketch two models of
approximately the same size to show the area that represents three-tenths is smaller than
the area that represents five-tenths.)


Students treat decimals as whole numbers when making comparison of two decimals. They think
the longer the number, the greater the value. For example, they think that ).03 is greater than 0.3.

Measurement and Data
CLUSTER #1: SOLVE PROBLEMS INVOLVING MEASUREMENT AND
CONVERSION OF MEASUREMENTS FROM A LARGER UNIT TO A SMALLER
UNIT.
should learn to use with increasing precision with this cluster are: measure, metric,
customary, convert/conversion, relative size, liquid volume, mass, length, distance,
kilometer (km), meter (m), centimeter (cm), kilogram (kg), gram (g), liter (L), milliliter
(mL), inch (in), foot (ft), yard (yd), mile (mi), ounce (oz), pound (lb), cup (c), pint (pt),
quart (qt), gallon (gal), time, hour, minute, second, equivalent, operations, add,
subtract, multiply, divide, fractions, decimals, area, perimeter.
MCC.4.MD.1 Know relative sizes of measurement units within one system of units
including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of
measurement, express measurements in a larger unit in terms of a smaller unit. Record
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measurement equivalents in a two-column table. For example, know that 1 ft is 12 times as
long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet
and inches listing the number pairs (1, 12), (2, 24), (3, 36), ...
The units of measure that have not been addressed in prior years are cups, pints, quarts,
gallons, pounds, ounces, kilometers, milliliters, and seconds. Students’ prior experiences
were limited to measuring length, mass (metric and customary systems), liquid volume
(metric only), and elapsed time. Students did not convert measurements. Students need
ample opportunities to become familiar with these new units of measure and explore the
patterns and relationships in the conversion tables that they create.
Students may use a two-column chart to convert from larger to smaller units and record
equivalent measurements. They make statements such as, if one foot is 12 inches, then 3
feet has to be 36 inches because there are 3 groups of 12.
Example:
Customary length conversion table
Yards Feet
1 3
2 6
3 9
n n×3
Foundational understandings to help with measure concepts:
• Understand that larger units can be subdivided into equivalent units (partition).
• Understand that the same unit can be repeated to determine the measure
(iteration).
Understand the relationship between the size of a unit and the number of units needed
(compensatory principle2).

MCC.4.MD.2 Use the four operations to solve word problems involving distances, intervals
of time, liquid volumes, masses of objects, and money, including problems involving simple
fractions or decimals, and problems that require expressing measurements given in a
larger unit in terms of a smaller unit. Represent measurement quantities using diagrams
such as number line diagrams that feature a measurement scale.

This standard includes multi-step word problems related to expressing measurements
from a larger unit in terms of a smaller unit (e.g., feet to inches, meters to centimeter,
dollars to cents). Students should have ample opportunities to use number line diagrams
to solve word problems.

Example:
Charlie and 10 friends are planning for a pizza party. They purchased 3 quarts of milk. If
each glass holds 8oz will everyone get at least one glass of milk?
4
The compensatory principle states that the smaller the unit used to measure the distance, the more of those units that will be needed. For example,
measuring a distance in centimeters will result in a larger number of that unit than measuring the distance in meters.

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Possible solution: Charlie plus 10 friends = 11 total people
11 people × 8 ounces (glass of milk) = 88 total ounces
1 quart = 2 pints = 4 cups = 32 ounces
Therefore 1 quart = 2 pints = 4 cups = 32 ounces
2 quarts = 4 pints = 8 cups = 64 ounces
3 quarts = 6 pints = 12 cups = 96 ounces

If Charlie purchased 3 quarts (6 pints) of milk there would be enough for everyone at his
party to have at least one glass of milk. If each person drank 1 glass then he would have
1- 8 oz glass or 1 cup of milk left over.

Additional examples with various operations:
• Division/fractions: Susan has 2 feet of ribbon. She wants to give her ribbon to her
3 best friends so each friend gets the same amount. How much ribbon will each friend
get?

Students may record their solutions using fractions or inches. (The answer would be 2/3
of a foot or 8 inches. Students are able to express the answer in inches because they
understand that 1/3 of a foot is 4 inches and 2/3 of a foot is 2 groups of 1/3.)
• Addition: Mason ran for an hour and 15 minutes on Monday, 25 minutes on
Tuesday, and 40 minutes on Wednesday. What was the total number of minutes Mason
ran?
• Subtraction: A pound of apples costs $1.20. Rachel bought a pound and a half of
apples. If she gave the clerk a $5.00 bill, how much change will she get back?
• Multiplication: Mario and his 2 brothers are selling lemonade. Mario brought one
and a half liters, Javier brought 2 liters, and Ernesto brought 450 milliliters. How many
total milliliters of lemonade did the boys have?

Number line diagrams that feature a measurement scale can represent measurement
quantities. Examples include: ruler, diagram marking off distance along a road with
cities at various points, a timetable showing hours throughout the day, or a volume
measure on the side of a container.

Example:
At 7:00 a.m. Candace wakes up to go to school. It takes her 8 minutes to shower, 9
minutes to get dressed and 17 minutes to eat breakfast. How many minutes does she have
until the bus comes at 8:00 a.m.? Use the number line to help solve the problem.

MCC.4.MD.3 Apply the area and perimeter formulas for rectangles in real world and
mathematical problems. For example, find the width of a rectangular room given the area of
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the flooring and the length, by viewing the area formula as a multiplication equation with an
unknown factor.

Students developed understanding of area and perimeter in 3rd grade by using visual
models.
While students are expected to use formulas to calculate area and perimeter of rectangles,
they need to understand and be able to communicate their understanding of why the
formulas work. The formula for area is I x w and the answer will always be in square
units. The formula for perimeter can be 2 l + 2 w or 2 (l + w) and the answer will be in
linear units. This standard calls for students to generalize their understanding of area and
perimeter by connecting the concepts to mathematical formulas. These formulas should
be developed through experience not just memorization.
Example:
Mr. Rutherford is covering the miniature golf course with an artificial grass. How
many 1-foot squares of carpet will he need to cover the entire course?

Students believe that larger units will give the larger measure. Students should be given multiple
opportunities to measure the same object with different measuring units. For example, have the
students measure the length of a room with one-inch tiles, with one-foot rulers, and with yard
sticks. Students should notice that it takes fewer yard sticks to measure the room than rulers or
tiles.

CLUSTER #2: REPRESENT AND INTERPRET DATA.
should learn to use with increasing precision with this cluster are: data, line plot, length,
fractions.
MCC.4.MD.4 Make a line plot to display a data set of measurements in fractions of a unit
(1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using
information presented in line plots. For example, from a line plot find and interpret the
difference in length between the longest and shortest specimens in an insect collection.
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This standard provides a context for students to work with fractions by measuring objects
to an eighth of an inch. Students are making a line plot of this data and then adding and
subtracting fractions based on data in the line plot.
Example:
Students measured objects in their desk to the nearest 1/2, 1/4, or 1/8 inch. They
displayed their data collected on a line plot. How many objects measured 1/4 inch? 1/2
inch? If you put all the objects together end to end what would be the total length of all
the objects.


Students use whole-number names when counting fractional parts on a number line. The fraction
name should be used instead. For example, if two-fourths is represented on the line plot three
times, then there would be six-fourths.

CLUSTER #3: GEOMETRIC MEASUREMENT – UNDERSTAND CONCEPTS OF
ANGLE AND MEASURE ANGLES.
should learn to use with increasing precision with this cluster are: measure, point, end
point, geometric shapes, ray, angle, circle, fraction, intersect, one-degree angle,
protractor, decomposed, addition, subtraction, unknown.
MCC.4.MD.5 Recognize angles as geometric shapes that are formed wherever two rays
share a common endpoint, and understand concepts of angle measurement:
This standard brings up a connection between angles and circular measurement (360
degrees).

a. An angle is measured with reference to a circle with its center at the common endpoint
of the rays, by considering the fraction of the circular arc between the points where the two
rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-
degree angle,” and can be used to measure angles.
The diagram below will help students understand that an angle measurement is not
related to an area since the area between the 2 rays is different for both circles yet the
angle measure is the same.

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Gr. 4 Overview

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